package ikor.math;

/** 
 * Singular Value Decomposition, 
 * adapted from JAMA (http://math.nist.gov/javanumerics/jama/)
 * and derived from LINPACK code (http://www.netlib.org/linpack/).
 * <p>
 *     From the JAMA documentation:
 * <p>
 *     For an m-by-n matrix A with m >= n, the singular value decomposition is
 *     an m-by-n orthogonal matrix U, an n-by-n diagonal matrix S, and
 *     an n-by-n orthogonal matrix V so that A = U*S*V'.
 * <p>
 *     The singular values, sigma[k] = S[k][k], are ordered so that
 *     sigma[0] >= sigma[1] >= ... >= sigma[n-1].
 * <p>
 *     The singular value decompostion always exists, so the constructor will
 *     never fail.  The matrix condition number and the effective numerical
 *     rank can be computed from this decomposition.
 */

public class SingularValueDecomposition extends MatrixDecomposition 
{
	// Internal data storage (JAMA implementation)
	
	/** Arrays for internal storage of U and V. */
	private double[][] U, V;

	/** Array for internal storage of singular values. */
	private double[] s;
	
	/** Array for internal storage of super-diagonal elements. */
	private double[] e;

	/** Row and column dimensions.  */
	private int m, n;

	// Constructor
	// -----------

	/** 
	 * Construct the singular value decomposition (U, S, V).
	 * 
	 * @param matrix Rectangular matrix
	 */

	public SingularValueDecomposition (Matrix matrix) 
	{
		m = matrix.rows();
		n = matrix.columns();

		int nu = Math.min(m,n);

		s = new double [Math.min(m+1,n)];
		U = new double [m][nu];
		V = new double [n][n];
		e = new double [n];

		int nct = Math.min(m-1,n);
		int nrt = Math.max(0,Math.min(n-2,m));

		// Set up the bidiagonal matrix or order p.

		double[][] A = matrix.getArray();
		double[] work = new double [m];

		// Reduce A to bidiagonal form, 
		// storing the diagonal elements
		// in s and the super-diagonal elements in e.

		for (int k = 0; k < Math.max(nct,nrt); k++) {
			if (k < nct) {

				// Compute the transformation for the k-th column and
				// place the k-th diagonal in s[k].
				// Compute 2-norm of k-th column without under/overflow.
				s[k] = 0;
				for (int i = k; i < m; i++) {
					s[k] = hypot(s[k],A[i][k]);
				}
				if (s[k] != 0.0) {
					if (A[k][k] < 0.0) {
						s[k] = -s[k];
					}
					for (int i = k; i < m; i++) {
						A[i][k] /= s[k];
					}
					A[k][k] += 1.0;
				}
				s[k] = -s[k];
			}
			for (int j = k+1; j < n; j++) {
				if ((k < nct) & (s[k] != 0.0))  {

					// Apply the transformation.

					double t = 0;
					for (int i = k; i < m; i++) {
						t += A[i][k]*A[i][j];
					}
					t = -t/A[k][k];
					for (int i = k; i < m; i++) {
						A[i][j] += t*A[i][k];
					}
				}

				// Place the k-th row of A into e for the
				// subsequent calculation of the row transformation.

				e[j] = A[k][j];
			}
			if (k < nct) {

				// Place the transformation in U for subsequent back
				// multiplication.

				for (int i = k; i < m; i++) {
					U[i][k] = A[i][k];
				}
			}
			if (k < nrt) {

				// Compute the k-th row transformation and place the
				// k-th super-diagonal in e[k].
				// Compute 2-norm without under/overflow.
				e[k] = 0;
				for (int i = k+1; i < n; i++) {
					e[k] = hypot(e[k],e[i]);
				}
				if (e[k] != 0.0) {
					if (e[k+1] < 0.0) {
						e[k] = -e[k];
					}
					for (int i = k+1; i < n; i++) {
						e[i] /= e[k];
					}
					e[k+1] += 1.0;
				}
				e[k] = -e[k];
				if ((k+1 < m) & (e[k] != 0.0)) {

					// Apply the transformation.

					for (int i = k+1; i < m; i++) {
						work[i] = 0.0;
					}
					for (int j = k+1; j < n; j++) {
						for (int i = k+1; i < m; i++) {
							work[i] += e[j]*A[i][j];
						}
					}
					for (int j = k+1; j < n; j++) {
						double t = -e[j]/e[k+1];
						for (int i = k+1; i < m; i++) {
							A[i][j] += t*work[i];
						}
					}
				}

				// Place the transformation in V for subsequent
				// back multiplication.

				for (int i = k+1; i < n; i++) {
					V[i][k] = e[i];
				}
			}
		}

		// Set up the final bidiagonal matrix or order p.

		int p = Math.min(n,m+1);
		if (nct < n) {
			s[nct] = A[nct][nct];
		}
		if (m < p) {
			s[p-1] = 0.0;
		}
		if (nrt+1 < p) {
			e[nrt] = A[nrt][p-1];
		}
		e[p-1] = 0.0;

		// Generate U.

		generateU(nu, nct);

		// Generate V.

		generateV(nu,nrt);

		// Main iteration loop for the singular values.

		int pp = p-1;
		int iter = 0;
		double eps = Math.pow(2.0,-52.0);
		double tiny = Math.pow(2.0,-966.0);
		while (p > 0) {
			int k,kase;

			// Here is where a test for too many iterations would go.

			// This section of the program inspects for
			// negligible elements in the s and e arrays.  On
			// completion the variables kase and k are set as follows.

			// kase = 1     if s(p) and e[k-1] are negligible and k<p
			// kase = 2     if s(k) is negligible and k<p
			// kase = 3     if e[k-1] is negligible, k<p, and
			//              s(k), ..., s(p) are not negligible (qr step).
			// kase = 4     if e(p-1) is negligible (convergence).

			for (k = p-2; k >= -1; k--) {
				if (k == -1) {
					break;
				}
				if (Math.abs(e[k]) <=
						tiny + eps*(Math.abs(s[k]) + Math.abs(s[k+1]))) {
					e[k] = 0.0;
					break;
				}
			}
			if (k == p-2) {
				kase = 4;
			} else {
				int ks;
				for (ks = p-1; ks >= k; ks--) {
					if (ks == k) {
						break;
					}
					double t = (ks != p ? Math.abs(e[ks]) : 0.) + 
							(ks != k+1 ? Math.abs(e[ks-1]) : 0.);
					if (Math.abs(s[ks]) <= tiny + eps*t)  {
						s[ks] = 0.0;
						break;
					}
				}
				if (ks == k) {
					kase = 3;
				} else if (ks == p-1) {
					kase = 1;
				} else {
					kase = 2;
					k = ks;
				}
			}
			k++;

			// Perform the task indicated by kase.

			switch (kase) {

			// Deflate negligible s(p).

			case 1: 
				deflateNegligibleSingularValue(p, k);
				break;

				// Split at negligible s(k).

			case 2: 
				splitNegligibleSingularValue(pp, k);
				break;

				// Perform one qr step.

			case 3: 
				e[p-2] = qr(p,k);
				iter = iter + 1;
				break;

				// Convergence.

			case 4: 
				convergence(pp,k);
				iter = 0;
				p--;
				break;
			}
		}
	}


	// Public methods
	// --------------
	
	/** 
	 * Return the left singular vectors
	 * @return U
	 */

	public Matrix getU () 
	{
		return new DenseMatrix(U); // m x Math.min(m+1,n);
	}

	/** 
	 * Return the right singular vectors
	 * @return V
	 */

	public Matrix getV () 
	{
		return new DenseMatrix(V); // n x n
	}

	/** 
	 * Return the vector of singular values
	 * @return diagonal of S.
	 */

	public Vector getSingularValues () 
	{
		return new DenseVector(s);
	}

	/** 
	 * Return the diagonal matrix of singular values
	 * @return S
	 */

	public Matrix getS () 
	{
		Matrix S = new DenseMatrix(n,n);
		
		for (int i = 0; i < n; i++) 
			S.set(i,i, this.s[i]);
		
		return S;
	}
	
	// Private methods
	// ---------------

	/** 
	 * Two norm
	 * @return max(S)
	 */

	public double norm2 () 
	{
		return s[0];
	}

	/** 
	 * Two norm condition number
	 * @return max(S)/min(S)
	 */

	public double cond () 
	{
		return s[0]/s[Math.min(m,n)-1];
	}

	/** 
	 * Effective numerical matrix rank
	 * @return Number of nonnegligible singular values.
	 */

	public int rank () 
	{
		double eps = Math.pow(2.0,-52.0);
		double tol = Math.max(m,n)*s[0]*eps;
		int r = 0;
		for (int i = 0; i < s.length; i++) {
			if (s[i] > tol) {
				r++;
			}
		}
		return r;
	}
	
	// Private methods
	// ---------------		

	// Generate V

	private void generateV (int nu, int nrt) 
	{
		for (int k = n-1; k >= 0; k--) {
			if ((k < nrt) & (e[k] != 0.0)) {
				for (int j = k+1; j < nu; j++) {
					double t = 0;
					for (int i = k+1; i < n; i++) {
						t += V[i][k]*V[i][j];
					}
					t = -t/V[k+1][k];
					for (int i = k+1; i < n; i++) {
						V[i][j] += t*V[i][k];
					}
				}
			}
			for (int i = 0; i < n; i++) {
				V[i][k] = 0.0;
			}
			V[k][k] = 1.0;
		}
	}

	private void generateU (int nu, int nct) 
	{
		for (int j = nct; j < nu; j++) {
			for (int i = 0; i < m; i++) {
				U[i][j] = 0.0;
			}
			U[j][j] = 1.0;
		}
		for (int k = nct-1; k >= 0; k--) {
			if (s[k] != 0.0) {
				for (int j = k+1; j < nu; j++) {
					double t = 0;
					for (int i = k; i < m; i++) {
						t += U[i][k]*U[i][j];
					}
					t = -t/U[k][k];
					for (int i = k; i < m; i++) {
						U[i][j] += t*U[i][k];
					}
				}
				for (int i = k; i < m; i++ ) {
					U[i][k] = -U[i][k];
				}
				U[k][k] = 1.0 + U[k][k];
				for (int i = 0; i < k-1; i++) {
					U[i][k] = 0.0;
				}
			} else {
				for (int i = 0; i < m; i++) {
					U[i][k] = 0.0;
				}
				U[k][k] = 1.0;
			}
		}
	}
	
	// Convergence
	
	private int convergence (int pp, int k) 
	{
		// Make the singular values positive.

		if (s[k] <= 0.0) {
			s[k] = (s[k] < 0.0 ? -s[k] : 0.0);
			for (int i = 0; i <= pp; i++) {
				V[i][k] = -V[i][k];
			}
		}

		// Order the singular values.

		while (k < pp) {
			if (s[k] >= s[k+1]) {
				break;
			}
			double t = s[k];
			s[k] = s[k+1];
			s[k+1] = t;
			if (k < n-1) {
				for (int i = 0; i < n; i++) {
					t = V[i][k+1]; V[i][k+1] = V[i][k]; V[i][k] = t;
				}
			}
			if (k < m-1) {
				for (int i = 0; i < m; i++) {
					t = U[i][k+1]; U[i][k+1] = U[i][k]; U[i][k] = t;
				}
			}
			k++;
		}
		return k;
	}

	// qr step
	
	private double qr (int p, int k) 
	{
		// Calculate the shift.

		double scale = Math.max(Math.max(Math.max(Math.max(
				Math.abs(s[p-1]),Math.abs(s[p-2])),Math.abs(e[p-2])), 
				Math.abs(s[k])),Math.abs(e[k]));
		double sp = s[p-1]/scale;
		double spm1 = s[p-2]/scale;
		double epm1 = e[p-2]/scale;
		double sk = s[k]/scale;
		double ek = e[k]/scale;
		double b = ((spm1 + sp)*(spm1 - sp) + epm1*epm1)/2.0;
		double c = (sp*epm1)*(sp*epm1);
		double shift = 0.0;
		if ((b != 0.0) | (c != 0.0)) {
			shift = Math.sqrt(b*b + c);
			if (b < 0.0) {
				shift = -shift;
			}
			shift = c/(b + shift);
		}
		double f = (sk + sp)*(sk - sp) + shift;
		double g = sk*ek;

		// Chase zeros.

		for (int j = k; j < p-1; j++) {
			double t = hypot(f,g);
			double cs = f/t;
			double sn = g/t;
			if (j != k) {
				e[j-1] = t;
			}
			f = cs*s[j] + sn*e[j];
			e[j] = cs*e[j] - sn*s[j];
			g = sn*s[j+1];
			s[j+1] = cs*s[j+1];

			// V
			for (int i = 0; i < n; i++) {
				t = cs*V[i][j] + sn*V[i][j+1];
				V[i][j+1] = -sn*V[i][j] + cs*V[i][j+1];
				V[i][j] = t;
			}

			t = hypot(f,g);
			cs = f/t;
			sn = g/t;
			s[j] = t;
			f = cs*e[j] + sn*s[j+1];
			s[j+1] = -sn*e[j] + cs*s[j+1];
			g = sn*e[j+1];
			e[j+1] = cs*e[j+1];
			
			// U
			if (j < m-1) {
				for (int i = 0; i < m; i++) {
					t = cs*U[i][j] + sn*U[i][j+1];
					U[i][j+1] = -sn*U[i][j] + cs*U[i][j+1];
					U[i][j] = t;
				}
			}
		}
		return f;
	}

	// Split at negligible s(k)
	
	private void splitNegligibleSingularValue (int p, int k) 
	{
		double f = e[k-1];
		e[k-1] = 0.0;
	
		for (int j = k; j < p; j++) {
			double t = hypot(s[j],f);
			double cs = s[j]/t;
			double sn = f/t;
			s[j] = t;
			f = -sn*e[j];
			e[j] = cs*e[j];
		
			// U
			
			for (int i = 0; i < m; i++) {
				t = cs*U[i][j] + sn*U[i][k-1];
				U[i][k-1] = -sn*U[i][j] + cs*U[i][k-1];
				U[i][j] = t;
			}
		}
	}

	// Deflate negligible s(k)
	
	private void deflateNegligibleSingularValue (int p, int k) 
	{
		double f = e[p-2];
		e[p-2] = 0.0;

		for (int j = p-2; j >= k; j--) {
			double t = hypot(s[j],f);
			double cs = s[j]/t;
			double sn = f/t;
			s[j] = t;
			if (j != k) {
				f = -sn*e[j-1];
				e[j-1] = cs*e[j-1];
			}
		
			// V
			
			for (int i = 0; i < n; i++) {
				t = cs*V[i][j] + sn*V[i][p-1];
				V[i][p-1] = -sn*V[i][j] + cs*V[i][p-1];
				V[i][j] = t;
			}
		}
	}
	
}
